How is possible that those shapes are equivalent in topology?
$begingroup$
I recently started to study topology, I have no idea about the subject so my question could be very simple but I need a clear explanation. It is about the page number 19 of Introducton to Topology by Colin Adams and Robert Franzosa; it said that the shapes:
are equivalent in topology, but one has just one hole and the other has two. is possible to add holes or stick holes?
general-topology
edited Aug 25 '18 at 19:05
Micah
29.8k1364106
asked Aug 25 '18 at 19:00
José MarínJosé Marín
13417
$endgroup$
add a comment |
$begingroup$
I recently started to study topology, I have no idea about the subject so my question could be very simple but I need a clear explanation. It is about the page number 19 of Introducton to Topology by Colin Adams and Robert Franzosa; it said that the shapes:
are equivalent in topology, but one has just one hole and the other has two. is possible to add holes or stick holes?
general-topology
edited Aug 25 '18 at 19:05
Micah
29.8k1364106
asked Aug 25 '18 at 19:00
José MarínJosé Marín
13417
$endgroup$
1
$begingroup$
Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges.
$endgroup$
– Michael Hardy
Aug 26 '18 at 1:27
4
$begingroup$
It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid.
$endgroup$
– user247327
Aug 26 '18 at 3:31
1
$begingroup$
The drawing is perhaps not the best but hopefully the others explained it.
$endgroup$
– Tom
Aug 27 '18 at 9:50
add a comment |
$begingroup$
I recently started to study topology, I have no idea about the subject so my question could be very simple but I need a clear explanation. It is about the page number 19 of Introducton to Topology by Colin Adams and Robert Franzosa; it said that the shapes:
are equivalent in topology, but one has just one hole and the other has two. is possible to add holes or stick holes?
general-topology
edited Aug 25 '18 at 19:05
Micah
29.8k1364106
asked Aug 25 '18 at 19:00
José MarínJosé Marín
13417
$endgroup$
I recently started to study topology, I have no idea about the subject so my question could be very simple but I need a clear explanation. It is about the page number 19 of Introducton to Topology by Colin Adams and Robert Franzosa; it said that the shapes:
are equivalent in topology, but one has just one hole and the other has two. is possible to add holes or stick holes?
general-topology
general-topology
edited Aug 25 '18 at 19:05
Micah
29.8k1364106
asked Aug 25 '18 at 19:00
José MarínJosé Marín
13417
edited Aug 25 '18 at 19:05
Micah
29.8k1364106
asked Aug 25 '18 at 19:00
José MarínJosé Marín
13417
edited Aug 25 '18 at 19:05
Micah
29.8k1364106
edited Aug 25 '18 at 19:05
Micah
29.8k1364106
edited Aug 25 '18 at 19:05
Micah
29.8k1364106
29.8k1364106
asked Aug 25 '18 at 19:00
José MarínJosé Marín
13417
asked Aug 25 '18 at 19:00
José MarínJosé Marín
13417
asked Aug 25 '18 at 19:00
José MarínJosé Marín
13417
13417
1
$begingroup$
Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges.
$endgroup$
– Michael Hardy
Aug 26 '18 at 1:27
4
$begingroup$
It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid.
$endgroup$
– user247327
Aug 26 '18 at 3:31
1
$begingroup$
The drawing is perhaps not the best but hopefully the others explained it.
$endgroup$
– Tom
Aug 27 '18 at 9:50
add a comment |
1
$begingroup$
Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges.
$endgroup$
– Michael Hardy
Aug 26 '18 at 1:27
4
$begingroup$
It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid.
$endgroup$
– user247327
Aug 26 '18 at 3:31
1
$begingroup$
The drawing is perhaps not the best but hopefully the others explained it.
$endgroup$
– Tom
Aug 27 '18 at 9:50
1
1
$begingroup$
Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges.
$endgroup$
– Michael Hardy
Aug 26 '18 at 1:27
$begingroup$
Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges.
$endgroup$
– Michael Hardy
Aug 26 '18 at 1:27
4
4
$begingroup$
It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid.
$endgroup$
– user247327
Aug 26 '18 at 3:31
$begingroup$
It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid.
$endgroup$
– user247327
Aug 26 '18 at 3:31
1
1
$begingroup$
The drawing is perhaps not the best but hopefully the others explained it.
$endgroup$
– Tom
Aug 27 '18 at 9:50
$begingroup$
The drawing is perhaps not the best but hopefully the others explained it.
$endgroup$
– Tom
Aug 27 '18 at 9:50
add a comment |
4 Answers
4
active
oldest
votes
$begingroup$
Look a bit more closely at the second picture. There's a couple of little dotted lines connecting the two holes that may be a bit hard to see.
That is meant to convey the impression they are the two ends of a single, long, curved hole through the interior.
answered Aug 26 '18 at 0:34
The_SympathizerThe_Sympathizer
7,4852245
$endgroup$
29
$begingroup$
I see the trench, but which one is the superlaser and which one is the exhaust port?
$endgroup$
– Lamar Latrell
Aug 26 '18 at 4:39
add a comment |
$begingroup$
The "two holes" in that sphere are two ends of the same hole. (That is, if you drilled one hole all the way through a sphere, you would end up with something that looked very much like your picture.)
answered Aug 25 '18 at 19:04
MicahMicah
29.8k1364106
$endgroup$
3
$begingroup$
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
$endgroup$
– Antoni Parellada
Aug 25 '18 at 19:07
4
$begingroup$
It’s probably worth noting that in topology a “hole” is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
$endgroup$
– Fogmeister
Aug 26 '18 at 6:47
add a comment |
$begingroup$
You may also notice the tunel, which I agree with you it is not clear in this photo.
answered Aug 25 '18 at 19:10
dmtridmtri
1,4522521
$endgroup$
7
$begingroup$
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
$endgroup$
– Andreas Blass
Aug 25 '18 at 20:28
$begingroup$
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
$endgroup$
– dmtri
Aug 26 '18 at 6:10
add a comment |
$begingroup$
The cuboid and the sphere are topological euvivalent. Drill a hole through each body as indicated by the arrow. The resulting bodies are still topological equivalent.
answered Aug 25 '18 at 20:06
miracle173miracle173
7,33322247
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2894411%2fhow-is-possible-that-those-shapes-are-equivalent-in-topology%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Look a bit more closely at the second picture. There's a couple of little dotted lines connecting the two holes that may be a bit hard to see.
That is meant to convey the impression they are the two ends of a single, long, curved hole through the interior.
answered Aug 26 '18 at 0:34
The_SympathizerThe_Sympathizer
7,4852245
$endgroup$
29
$begingroup$
I see the trench, but which one is the superlaser and which one is the exhaust port?
$endgroup$
– Lamar Latrell
Aug 26 '18 at 4:39
add a comment |
$begingroup$
Look a bit more closely at the second picture. There's a couple of little dotted lines connecting the two holes that may be a bit hard to see.
That is meant to convey the impression they are the two ends of a single, long, curved hole through the interior.
answered Aug 26 '18 at 0:34
The_SympathizerThe_Sympathizer
7,4852245
$endgroup$
29
$begingroup$
I see the trench, but which one is the superlaser and which one is the exhaust port?
$endgroup$
– Lamar Latrell
Aug 26 '18 at 4:39
add a comment |
$begingroup$
Look a bit more closely at the second picture. There's a couple of little dotted lines connecting the two holes that may be a bit hard to see.
That is meant to convey the impression they are the two ends of a single, long, curved hole through the interior.
answered Aug 26 '18 at 0:34
The_SympathizerThe_Sympathizer
7,4852245
$endgroup$
Look a bit more closely at the second picture. There's a couple of little dotted lines connecting the two holes that may be a bit hard to see.
That is meant to convey the impression they are the two ends of a single, long, curved hole through the interior.
answered Aug 26 '18 at 0:34
The_SympathizerThe_Sympathizer
7,4852245
answered Aug 26 '18 at 0:34
The_SympathizerThe_Sympathizer
7,4852245
answered Aug 26 '18 at 0:34
The_SympathizerThe_Sympathizer
7,4852245
answered Aug 26 '18 at 0:34
The_SympathizerThe_Sympathizer
7,4852245
7,4852245
29
$begingroup$
I see the trench, but which one is the superlaser and which one is the exhaust port?
$endgroup$
– Lamar Latrell
Aug 26 '18 at 4:39
add a comment |
29
$begingroup$
I see the trench, but which one is the superlaser and which one is the exhaust port?
$endgroup$
– Lamar Latrell
Aug 26 '18 at 4:39
29
29
$begingroup$
I see the trench, but which one is the superlaser and which one is the exhaust port?
$endgroup$
– Lamar Latrell
Aug 26 '18 at 4:39
$begingroup$
I see the trench, but which one is the superlaser and which one is the exhaust port?
$endgroup$
– Lamar Latrell
Aug 26 '18 at 4:39
add a comment |
$begingroup$
The "two holes" in that sphere are two ends of the same hole. (That is, if you drilled one hole all the way through a sphere, you would end up with something that looked very much like your picture.)
answered Aug 25 '18 at 19:04
MicahMicah
29.8k1364106
$endgroup$
3
$begingroup$
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
$endgroup$
– Antoni Parellada
Aug 25 '18 at 19:07
4
$begingroup$
It’s probably worth noting that in topology a “hole” is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
$endgroup$
– Fogmeister
Aug 26 '18 at 6:47
add a comment |
$begingroup$
The "two holes" in that sphere are two ends of the same hole. (That is, if you drilled one hole all the way through a sphere, you would end up with something that looked very much like your picture.)
answered Aug 25 '18 at 19:04
MicahMicah
29.8k1364106
$endgroup$
3
$begingroup$
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
$endgroup$
– Antoni Parellada
Aug 25 '18 at 19:07
4
$begingroup$
It’s probably worth noting that in topology a “hole” is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
$endgroup$
– Fogmeister
Aug 26 '18 at 6:47
add a comment |
$begingroup$
The "two holes" in that sphere are two ends of the same hole. (That is, if you drilled one hole all the way through a sphere, you would end up with something that looked very much like your picture.)
answered Aug 25 '18 at 19:04
MicahMicah
29.8k1364106
$endgroup$
The "two holes" in that sphere are two ends of the same hole. (That is, if you drilled one hole all the way through a sphere, you would end up with something that looked very much like your picture.)
answered Aug 25 '18 at 19:04
MicahMicah
29.8k1364106
answered Aug 25 '18 at 19:04
MicahMicah
29.8k1364106
answered Aug 25 '18 at 19:04
MicahMicah
29.8k1364106
answered Aug 25 '18 at 19:04
MicahMicah
29.8k1364106
29.8k1364106
3
$begingroup$
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
$endgroup$
– Antoni Parellada
Aug 25 '18 at 19:07
4
$begingroup$
It’s probably worth noting that in topology a “hole” is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
$endgroup$
– Fogmeister
Aug 26 '18 at 6:47
add a comment |
3
$begingroup$
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
$endgroup$
– Antoni Parellada
Aug 25 '18 at 19:07
4
$begingroup$
It’s probably worth noting that in topology a “hole” is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
$endgroup$
– Fogmeister
Aug 26 '18 at 6:47
3
3
$begingroup$
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
$endgroup$
– Antoni Parellada
Aug 25 '18 at 19:07
$begingroup$
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
$endgroup$
– Antoni Parellada
Aug 25 '18 at 19:07
4
4
$begingroup$
It’s probably worth noting that in topology a “hole” is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
$endgroup$
– Fogmeister
Aug 26 '18 at 6:47
$begingroup$
It’s probably worth noting that in topology a “hole” is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
$endgroup$
– Fogmeister
Aug 26 '18 at 6:47
add a comment |
$begingroup$
You may also notice the tunel, which I agree with you it is not clear in this photo.
answered Aug 25 '18 at 19:10
dmtridmtri
1,4522521
$endgroup$
7
$begingroup$
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
$endgroup$
– Andreas Blass
Aug 25 '18 at 20:28
$begingroup$
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
$endgroup$
– dmtri
Aug 26 '18 at 6:10
add a comment |
$begingroup$
You may also notice the tunel, which I agree with you it is not clear in this photo.
answered Aug 25 '18 at 19:10
dmtridmtri
1,4522521
$endgroup$
7
$begingroup$
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
$endgroup$
– Andreas Blass
Aug 25 '18 at 20:28
$begingroup$
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
$endgroup$
– dmtri
Aug 26 '18 at 6:10
add a comment |
$begingroup$
You may also notice the tunel, which I agree with you it is not clear in this photo.
answered Aug 25 '18 at 19:10
dmtridmtri
1,4522521
$endgroup$
You may also notice the tunel, which I agree with you it is not clear in this photo.
answered Aug 25 '18 at 19:10
dmtridmtri
1,4522521
answered Aug 25 '18 at 19:10
dmtridmtri
1,4522521
answered Aug 25 '18 at 19:10
dmtridmtri
1,4522521
answered Aug 25 '18 at 19:10
dmtridmtri
1,4522521
1,4522521
7
$begingroup$
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
$endgroup$
– Andreas Blass
Aug 25 '18 at 20:28
$begingroup$
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
$endgroup$
– dmtri
Aug 26 '18 at 6:10
add a comment |
7
$begingroup$
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
$endgroup$
– Andreas Blass
Aug 25 '18 at 20:28
$begingroup$
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
$endgroup$
– dmtri
Aug 26 '18 at 6:10
7
7
$begingroup$
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
$endgroup$
– Andreas Blass
Aug 25 '18 at 20:28
$begingroup$
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
$endgroup$
– Andreas Blass
Aug 25 '18 at 20:28
$begingroup$
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
$endgroup$
– dmtri
Aug 26 '18 at 6:10
$begingroup$
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
$endgroup$
– dmtri
Aug 26 '18 at 6:10
add a comment |
$begingroup$
The cuboid and the sphere are topological euvivalent. Drill a hole through each body as indicated by the arrow. The resulting bodies are still topological equivalent.
answered Aug 25 '18 at 20:06
miracle173miracle173
7,33322247
$endgroup$
add a comment |
$begingroup$
The cuboid and the sphere are topological euvivalent. Drill a hole through each body as indicated by the arrow. The resulting bodies are still topological equivalent.
answered Aug 25 '18 at 20:06
miracle173miracle173
7,33322247
$endgroup$
add a comment |
$begingroup$
The cuboid and the sphere are topological euvivalent. Drill a hole through each body as indicated by the arrow. The resulting bodies are still topological equivalent.
answered Aug 25 '18 at 20:06
miracle173miracle173
7,33322247
$endgroup$
The cuboid and the sphere are topological euvivalent. Drill a hole through each body as indicated by the arrow. The resulting bodies are still topological equivalent.
answered Aug 25 '18 at 20:06
miracle173miracle173
7,33322247
answered Aug 25 '18 at 20:06
miracle173miracle173
7,33322247
answered Aug 25 '18 at 20:06
miracle173miracle173
7,33322247
answered Aug 25 '18 at 20:06
miracle173miracle173
7,33322247
7,33322247
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2894411%2fhow-is-possible-that-those-shapes-are-equivalent-in-topology%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges.
$endgroup$
– Michael Hardy
Aug 26 '18 at 1:27
4
$begingroup$
It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid.
$endgroup$
– user247327
Aug 26 '18 at 3:31
1
$begingroup$
The drawing is perhaps not the best but hopefully the others explained it.
$endgroup$
– Tom
Aug 27 '18 at 9:50